Mathematical Sciences Faculty Publications Mathematical Sciences, Department of
2015
A New Class of Generalized Power Lindley
Distribution with Applications to Lifetime Data
Marvis Pararai
Indiana University of Pennsylvania
Gayan Warahena-Liyanage
Indiana University of Pennsylvania
Broderick O. Oluyede
Georgia Southern University, [email protected]
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Recommended Citation
Pararai, Marvis, Gayan Warahena-Liyanage, Broderick O. Oluyede. 2015. "A New Class of Generalized Power Lindley Distribution with Applications to Lifetime Data."Theoretical Mathematics and Applications, 5 (1): 53-96. source: http://www.scienpress.com/ journal_focus.asp?main_id=60&Sub_id=IV&Issue=1374
Scienpress Ltd, 2015
A New Class of Generalized Power Lindley
Distribution with Applications to Lifetime Data
Mavis Pararai1, Gayan Warahena-Liyanage2 and Broderick O. Oluyede3
Abstract
A new class of distribution called the beta-exponentiated power Lindley (BEPL) distribution is proposed. This class of distributions includes the Lindley (L), exponentiated Lindley (EL), power Lindley (PL), exponentiated power Lindley (EPL), beta-exponentiated Lindley (BEL), beta-Lindley (BL), and beta-power Lindley distributions (BPL) as special cases. Expansion of the density of BEPL distribution is ob-tained. Some mathematical properties of the new distribution including hazard function, reverse hazard function, moments, mean deviations, Lorenz and Bonferroni curves are presented. Entropy measures and the distribution of the order statistics are given. The maximum likelihood estimation technique is used to estimate the model parameters. Fi-nally, real data examples are discussed to illustrate the usefulness and applicability of the proposed distribution.
Mathematics Subject Classification: 60E05; 62E15
1Department of Mathematics, Indiana University of Pennsylvania, Indiana, PA, 15705,
USA. E-mail: [email protected]
2Department of Mathematics, Indiana University of Pennsylvania, Indiana, PA, 15705,
USA. E-mail: [email protected]
3Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA,
30460,USA. E-mail: [email protected]
Article Info: Received : August 7, 2014. Revised : September 19, 2014.
Keywords: Exponentiated power Lindley distribution; Power Lindley distri-bution; Lindley distridistri-bution; Beta distridistri-bution; Maximum likelihood estima-tion
1
Introduction
Lindley [1] developed Lindley distribution in the context of fiducial and Bayesian statistics. Properties, extensions and applications of the Lindley distribution have been studied in the context of reliability analysis by Ghitany et al. [2], Zakerzadeh and Dolati [3], and Warahena-Liyanage and Pararai [4]. Several other authors including Sankaran [5], Asgharzadeh et al. [6] and Nadarajah et al. [7] proposed and developed the mathematical properties of various generalized Lindley distributions. The probability density function (pdf) of the Lindley distribution is given by
f(y;β) = β
2
β+ 1(1 +y)e
−βy, y >0, β > 0.
The power Lindley (PL) distribution proposed by Ghitany et al. [8] is an extension of the Lindley (L) distribution. Using the transformationX =Y α1,
Ghitany et al. [8] derived and studied the power Lindley (PL) distribution with the probability density function (pdf) given by
f(x;α, β) = αβ
2
β+ 1(1 +x
α)xα−1e−βxα, x >0, α >0, β >0.
The cumulative distribution function (cdf) of the power Lindley distribution is F(x) = 1−S(x) = 1− 1 + βx α β+ 1 e−βxα
forx >0, α, β >0.Warahena-Liyanage and Pararai [4] studied the properties of the exponentiated Power Lindley (EPL) distribution. The EPL cdf and pdf are given by GEP L(x;α, β, ω) = 1− 1 + βx α β+ 1 e−βxα ω (1.1)
and gEP L(x;α, β, ω) = αβ2ω β+ 1(1 +x α)xα−1e−βxα 1− 1 + βx α β+ 1 e−βxα ω−1 , (1.2) respectively, for x > 0, α > 0, β > 0, ω > 0. The hazard rate function of the EPL distribution is given by
hGEP L(x;α, β, ω) = g(x;α, β, ω) G(x;α, β, ω) = αβ2ω β+1(1 +x α)xα−1e−βxαh 1−1 + ββx+1αe−βxαiω−1 1− 1− 1 + βx α β+ 1 e−βxα ω .
The rth moment of the EPL distribution is given by
E(Xr) = ∞ X i=0 i X j=0 j+1 X k=0 ω−1 i i j j+ 1 k (−1)iβj−k−rα−1+1 Γ(k+rα−1+ 1) (β+ 1)i+1(i+ 1)(k+rα−1+1) .
The purpose of this paper is to develop a five-parameter alternative to several lifetime distributions including the gamma, Weibull, exponentiated Weibull, exponentiated Lindley, lognormal, beta Weibull geometric (BWG) [9], and beta Weibull Poisson (BWP) [10] distributions. In this context, we propose and develop the statistical properties of the beta exponentiated power Lindley (BEPL) distribution and show that it is a competitive model for reli-ability analysis. Our aim in this paper is to discuss some important statistical properties of the BEPL distribution. This discussion includes the shapes of the density, hazard rate and reverse hazard rate functions, moments, moment generating function and parameter estimation by using the method of maxi-mum likelihood. Finally, applications of the model to real data sets in order to illustrate the applicability and usefulness of the BEPL distribution are pre-sented.
This paper is organized as follows. In section 2,the model, sub-models and some of its statistical properties including shapes and behavior of the hazard function are presented. Moments, conditional moments, reliability and related measures are given in section 3.Mean deviations, Bonferroni and Lorenz curves are presented in section 4. Section 5 contains distribution of order statistics and measures of uncertainty. In section 6,we present the maximum likelihood
method for estimating the parameters of the distribution. Applications are given in section 7 followed by concluding remarks.
2
The Model, Sub-models and Some
Proper-ties
In this section, we present the BEPL distribution and derive some prop-erties of this class of distributions including the cdf, pdf, expansion of the density, hazard and reverse hazard functions, shape and sub-models. LetG(x) denote the cdf of a continuous random variable X and define a general class of distributions by F(x) = BG(x)(a, b) B(a, b) , (2.1) where BG(x)(a, b) = RG(x) 0 t a−1(1−t)b−1dt and 1/B(a, b) = Γ(a+b)/Γ(a)Γ(b).
The class of generalized distributions above was motivated by the work of Eu-gene et al. [11]. They proposed and studied the beta-normal distribution. Some beta-generalized distributions discussed in the literature include work by Jones [12], Bidram et al. [9]. Nadarajah and Kotz [13], Nadarajah and Gupta [14], Nadarajah and Kotz [15], Barreto-Souza et al. [16] proposed the beta-Gumbel, beta-Frechet, beta-exponential (BE), beta-exponentiated expo-nential (BEE) distributions, respectively. Gusmao et al. [17] presented results on the generalized inverse Weibull distribution. Pescim et al. [18] proposed and studied the beta-generalized half-normal distribution which contains some important distributions such as the half-normal and generalized half normal (Cooray and Ananda [19]) as special cases. Furthermore, Cordeiro et al. [20] presented the generalized Rayleigh distribution and Carrasco et al. [21] stud-ied the generalized modifstud-ied Weibull distribution with applications to lifetime data. More recently, Oluyede and Yang [22] studied the beta generalized Lind-ley distribution with applications.
By considering G(x) as the cdf of EPL distribution we obtain the beta-exponentiated power Lindley (BEPL) distribution with a broad class of dis-tributions that may be applicable in a wide range of day to day situations
including applications in medicine, reliability and ecology. The cdf and pdf of the five-parameter BEPL distribution are given by
FBEP L(x;α, β, ω, a, b) = 1 B(a, b) Z GEP L(x;α,β,ω) 0 ta−1(1−t)b−1dt = BG(x)(a, b) B(a, b) , (2.2) and fBEP L(x;α, β, ω, a, b) = 1 B(a, b)[GEP L(x)] a−1 [1−GEP L(x)] b−1 gEP L(x), = αβ 2ω B(a, b)(β+ 1)(1 +x α)xα−1e−βxα × 1− 1 + βx α β+ 1 e−βxα ωa−1 × 1− 1− 1 + βx α β+ 1 e−βxα ωb−1 , (2.3) respectively, for x > 0, α > 0, β > 0, ω > 0, a > 0, b > 0. Plots of the pdf of BEPL distribution for several combinations of values of α, β, ω, aand b are given in Figure 2.1. The plots indicate that the BEPL pdf can be decreasing or right skewed. The BEPL distribution has a positive asymmetry.
Figure 2.1: Plots of the PDF for different values of α, β, ω, aand b
2.1
Expansion of density
Forb >0 a real non-integer, we use the series representation (1−GEP L(x))b−1 = ∞ X i=0 b−1 i (−1)i[GEP L(x)] i , (2.4) where GEP L(x;α, β, ω) = 1− 1 + βx α β+ 1 e−βxα ω .
If a is an integer, from Equation (2.3) and the above expansion (2.4), we can rewrite the density of the BEPL distribution as
fBEP L(x;α, β, ω, a, b) = gEP L(x) B(a, b) ∞ X i=0 b−1 i (−1)i[GEP L(x)]a+i −1 (2.5) = αβ 2ω β+ 1(1 +x α)xα−1e−βxα 1− 1 + βx α β+ 1 e−βxα ω−1 × ∞ X i=0 (−1)i b−1 i B(a, b) 1− 1 + βx α β+ 1 e−βxα ω(a+i−1) = αβ 2ω β+ 1(1 +x α)xα−1e−βxα × ∞ X i=0 li 1− 1 + βx α β+ 1 e−βxα ω(a+i)−1 , (2.6)
where the coefficientsli are
li =li(a, b) =
(−1)i b−i1
B(a, b)
and P∞i=0li = 1,for x >0, α >0, β >0, ω >0, a >0, b >0.
Ifa is real non-integer, we can expand [GEP L(x)] a+i−1 as follows: [GEP L(x)] a+i−1 = {1−[1−GEP L(x)]} a+i−1 = ∞ X j=0 a+i−1 j (−1)j[1−GEP L(x)]j, with [1−GEP L(x)]j = j X k=0 j k (−1)k[GEP L(x)]k,
so that [GEP L(x)] a+i−1 = ∞ X j=0 j X k=0 a+i−1 j j k (−1)j+k[GEP L(x)] k . (2.7)
From Equations (2.5) and (2.7), the BEPL density can be rearranged in the form fBEP L(x;α, β, ω, a, b) = gEP L(x) ∞ X i,j=0 j X k=0 li,j,k[GEP L(x)] k (2.8) = αβ 2ω β+ 1(1 +x α)xα−1e−βxα ) × ∞ X i,j=0 j X k=0 li,j,k 1− 1 + βx α β+ 1 e−βxα ω(k+1)−1 ,
where the coefficientsli,j,k are
li,j,k =li,j,k(a, b) = (−1)i+j+k b−1 i a+i−1 j j k B(a, b) and P∞i,j=0Pj
k=0li,j,k = 1, for x >0, α >0, β >0, ω > 0, a >0, b > 0. Hence,
for any real non-integer, the BEPL density is given by three (two infinite and one finite) weighted power series sums of the baseline cdf GEP L(x). By
changingP∞j=0Pj k=0 to P∞ k=0 P∞ j=k in Equation (2.8), we obtain fBEP L(x;α, β, ω, a, b) = gEP L(x) ∞ X i,k=0 pi[GEP L(x)]k = αβ 2ω β+ 1(1 +x α )xα−1e−βxα × ∞ X i,k=0 pi 1− 1 + βx α β+ 1 e−βxα ω(k+1)−1 ,
where the coefficientpi is
pi =pi(a, b) = (−1)i b−1 i qk(a+i−1) B(a, b) , with qk=qk(a+i−1) = ∞ X j=k a+i−1 j j k (−1)j+k,
for x >0, α > 0, β >0, ω > 0, a >0, b >0, respectively. Note that the BEPL density is given by three infinite weighted power series sums of the baseline distribution function GEP L(x). When b > 0 is an integer, the index i in the
previous series representation stops at b−1.
2.2
Some sub-models of the BEPL distribution
Some sub-models of the BEPL distribution for selected values of the pa-rameters α, β, ω, aand b are presented in this section.
(1) a =b = 1
When a = b = 1, we obtain the exponentiated power Lindley (EPL) distribution whose cdf and pdf are given in (1.1) and (1.2), (Warahena-Liyanage and Pararai [4]).
(2) ω = 1
When ω= 1, we obtain the beta-power Lindley (BPL) distribution. The BPL cdf is given by FBP L(x;α, β, a, b) = 1 B(a, b) Z GP L(x;α,β) 0 ta−1(1−t)b−1dt
for x >0, α >0, β > 0, a >0, b >0. The corresponding pdf is given by
fBP L(x;α, β, a, b) = αβ2 B(a, b)(β+ 1)(1 +x α)xα−1e−βxα × 1− 1 + βx α β+ 1 e−βxα a−1 1 + βx α β+ 1 e−βxα b−1 for x >0, α >0, β > 0, a >0, b >0. (3) α = 1
When α = 1, we obtain beta-exponentiated Lindley (BEL) distribution (Oluyede and Yang [22]). The BEL cdf is given by
FBEL(x;β, ω, a, b) = 1 B(a, b) Z GEL(x;β,ω) 0 ta−1(1−t)b−1dt
for x >0, β >0, ω >0, a >0, b >0. The corresponding pdf is given by fBEL(x;β, ω, a, b) = β2ω B(a, b)(β+ 1)(1 +x)e −βx × 1− 1 + βx β+ 1 e−βx ωa−1 × 1− 1− 1 + βx β+ 1 e−βx ωb−1 for x >0, β >0, ω >0, a >0, b >0. (4) ω =α= 1
When ω = α = 1, we obtain beta-Lindley (BL) distribution (Oluyede and Yang [22]). The BL cdf and pdf are given by
FBL(x;β, a, b) = 1 B(a, b) Z GL(x;β,ω) 0 ta−1(1−t)b−1dt and fBL(x;β, a, b) = β2 B(a, b)(β+ 1)(1 +x)e −βx × 1− 1 + βx β+ 1 e−βx a−1 1 + βx β+ 1 e−βx b−1 , respectively, for x >0, β >0, ω >0, a >0, b >0. (5) ω =a=b = 1
When ω = a = b = 1, we obtain the power Lindley (PL) distribution (Ghitany et al. [8]). The PL cdf and pdf are respectively given by
FP L(x;α, β) = 1− 1 + βx α β+ 1 e−βxα and fP L(x;α, β) = β2 (β+ 1)(1 +x α)xα−1e−βxα for x >0, α >0, β > 0.
(6) α =a =b = 1
When α = a = b = 1, we obtain exponentiated-Lindley (EL) distribu-tion. The EL cdf is given by
FEL(x;β, ω) = 1− 1 + βx β+ 1 e−βx ω
for x >0, β >0, ω >0. The corresponding pdf is given by
fEL(x;β, ω) = β2ω (β+ 1)(1 +x)e −βx 1− 1 + βx β+ 1 e−βx ω−1 for x >0, β >0, ω >0. (7) α =ω =a=b = 1
When α =ω =a =b = 1, we obtain Lindley distribution. The Lindley cdf and pdf are respectively given by
FL(x;β) = 1− 1 + βx β+ 1 e−βx and fL(x;β) = β2 (β+ 1)(1 +x)e −βx for x >0, β >0. (8) ω =a= 1
When ω=a= 1, the cdf of BEPL distribution reduces to
FBP L(x;α, β, b) = 1− 1 + βx α β+ 1 e−βxα b
for x >0, α >0, β > 0, b >0. The corresponding pdf is
fBP L(x;α, β, b) = bαβ2 (β+ 1)(1 +x α)xα−1e−βxα 1 + βx α β+ 1 e−βxα b−1 for x >0, α >0, β > 0, b >0. (9) α =a = 1
When α=a= 1, the cdf of BEPL distribution reduces to
FBEL(x;α, β, b) = 1− 1− 1− 1 + βx β+ 1 e−βx ωb
for x >0, β >0, ω >0, b >0. The corresponding pdf is given by fBEL(x;β, ω, b) = bωβ2 (β+ 1)(1 +x)e −βx × 1− 1 + βx β+ 1 e−βx ω−1 × 1− 1− 1 + βx β+ 1 e−βx ωb−1 ,
for x > 0, β > 0, ω > 0, b > 0. This is the Kumaraswamy Lindley distribution with parameters β, ω and b.
(10) α =ω =a= 1
When α=ω=a= 1, the cdf of BEPL distribution reduces to
FBL(x;β, b) = 1− 1 + βx β+ 1 e−βx b
for x >0, β >0, b >0. The corresponding pdf is given by
fBL(x;β, b) = bβ2 (β+ 1)(1 +x)e −βx 1 + βx β+ 1 e−βx b−1 for x >0, β >0, b >0.
2.3
Hazard and Reverse Hazard Functions
The hazard and reverse hazard functions of the BEPL distribution are presented in this section. Graphs of these functions for selected values of parametersα, β, ω, aand b are also presented. The hazard and reverse hazard functions of the BEPL distribution are given respectively by
hBEP L(x;α, β, ω, a, b) = fBEP L(x;α, β, ω, a, b) FBEP L(x;α, β, ω, a, b) = gEP L(x) [GEP L(x)] a−1 [1−GEP L(x)]b −1 B(a, b)−BGEP L(x)(a, b) and
τBEP L(x;α, β, ω, a, b) = fBEP L(x;α, β, ω, a, b) FBEP L(x;α, β, ω, a, b) = gEP L(x) [GEP L(x)] a−1 [1−GEP L(x)]b −1 BGEP L(x)(a, b) ,
for x > 0, α > 0, β > 0, ω > 0, a > 0, b > 0, where GEP L(x) and gEP L(x) are
the cdf and pdf of the EPL distribution given by Equations (1.1) and (1.2), respectively. Plots of the hazard function for selected values of parameters
α, β, ω, a and b are given in Figures 2.2 and 2.3. The graphs of the hazard function for several combinations of the parameters represent various shapes including monotonically increasing, monotonically decreasing, bathtub and up-side down bathtub shapes. This attractive flexibility makes BEPL hazard rate function useful and suitable for non-monotone empirical hazard behaviors which are more likely to be encountered or observed in real life situations.
Figure 2.2: Plots of the hazard function for different values ofα, β, ω, a and b
2.4
Monotonicity Properties
The monotonicity properties of the BEPL distribution are discussed in this section. Let V(x) = GP L(x;α, β) = 1− 1 + βx α β+ 1 e−βxα.
Figure 2.3: Plots of the hazard function for different values ofα, β, ω, a and b
From Equation (2.3) we can rewrite the BEPL pdf as
fBEP L(x;α, β, ω, a, b) = αβ2ω B(a, b)(β+ 1)(1 +x α )xα−1e−βxα × [V(x)]ωa−1[1−Vω(x)]b−1 forx >0, α >0, β > 0, ω > 0, a >0, b >0. It follows that
logfBEP L(x) = log
αβ2ω B(a, b)(β+ 1)
+ log(1 +xα) + (α−1) log(x)−βxα
+ (ωa−1) logV(x) + (b−1) log [1−Vα(x)], (2.9) and dlogfBEP L(x) dx = αxα−1 1 +xα + α−1 x −αβx α−1 + (ωa−1)(1−V ω(x))−ω(b−1)Vω(x) V(x) [1−Vω(x)] V 0 (x). (2.10) SubstitutingV0(x) =dV(x)/dx= (αβ2/(β+ 1))(1 +xα)xα−1e−βxα into Equa-tion (2.10), we have dlogfBEP L(x) dx = αxα−1 1 +xα + α−1 x −αβx α−1+ αβ 2 β+ 1(1 +x α)xα−1e−βxα × (ωa−1)(1−Vω(x))−ω(b−1)Vω(x) V(x) [1−Vω(x)] .
Sinceα >0, β >0, ω >0, a >0 and b >0, we have V0(x) = dV(x) dx = αβ2 β+ 1(1 +x α)xα−1e−βxα >0,∀x >0. (2.11) Ifx−→0, then V(x) = 1− 1 + βx α β+ 1 e−βxα −→0. Ifx−→ ∞, then V(x) = 1− 1 + βx α β+ 1 e−βxα −→1.
Thus V(x) is monotonically increasing from 0 to 1. Note that, since 0 < V(x)<1,
0 < Vω(x) < 1,∀ω > 0,0 < 1−Vω(x) < 1,∀ω > 0 and V0(x) > 0, we have
V0(x)/V(x)[1−Vω(x)]>0.
Ifα61/2, ωa <1 and b >1. we obtain
dlogfBEP L(x) dx = αxα−1 1 +xα + α−1 x −αβx α−1 + (ωa−1)(1−V ω(x))−ω(b−1)Vω(x) V(x) [1−Vω(x)] V 0 (x)<0 (2.12) since [αxα−1/(1 +xα)] + [(α−1)/x] = [(2α−1)xα+ (α−1)]/x(1 +xα)<0, (ωa−1)(1−Vω(x))−ω(b−1)Vω(x)<0 andV0(x)/V(x)[1−Vω(x)]>0.
In this case, fBEP L(x;α, β, ω, a, b) is monotonically decreasing for all x.
If α > 1/2, fBEP L(x;α, β, ω, a, b) could attain a maximum, a minimum or a
point of inflection according to whether
d2logf BEP L(x) dx2 <0, d2logf BEP L(x) dx2 >0 or d2logf BEP L(x) dx2 = 0.
3
Moments, Conditional Moments and
Relia-bility
In this section, moments, conditional moments and reliability and related measures including coefficients of variation, skewness and kurtosis of the BEPL
distribution are presented. A table of values for mean, variance, coefficient of skewness (CS) and coefficient of kurtosis (CK) is also presented.
3.1
Moments
The rth moment of the BEPL distribution, denoted byµ0r is given by
µ0r =E(Xr) =
Z ∞
−∞
xrfBEP L(x)dx for r= 0,1,2, . . . .
In order to find the moments of the BEPL distribution, consider the following lemma. Lemma 3.1. Let L1(α, β, m, r) = Z ∞ 0 (1 +xα)xα+r−1 1− 1 + βx α β+ 1 e−βxα m−1 e−βxαdx, then L1(α, β, m, r) = ∞ X j=0 j X k=0 k+1 X l=0 m−1 j j k k+ 1 l (−1)jβkΓ(l+rα−1+ 1) α(β+ 1)j[β(j + 1)](l+rα−1+1).
Proof. Using the series expansion
(1−z)a−1 = ∞ X i=0 a−1 i (−1)izi, (3.1)
where|z |<1 and b >0 is a real non-integer, we have
L1(α, β, m, r) = ∞ X j=0 m−1 j (−1)j Z ∞ 0 1 +β(1 +xα) β+ 1 j e−jβxα(1 +xα)xα+r−1e−βxαdx = ∞ X j=0 m−1 j (−1)j (β+ 1)j j X k=0 j k βk Z ∞ 0 (1 +xα)k+1xα+r−1e(−jβxα−βxα)dx = ∞ X j=0 m−1 j j X k=0 j k k+1 X l=0 k+ 1 l (−1)jβk (β+ 1)j Z ∞ 0 xα+αl+r−1e(−jβxα−βxα)dx.
Now consider,
Z ∞
0
xα+αl+r−1e(−jβxα−βxα)dx. (3.2)
Letu=β(j + 1)xα,then dxdu =αβ(j+ 1)xα−1 and x=
u β(j+ 1) 1/α . Consequently, L1(α, β, m, r) = ∞ X j=0 j X k=0 k+1 X l=0 m−1 j j k k+ 1 l (−1)jβkΓ(l+rα−1+ 1) α(β+ 1)j[β(j + 1)](l+rα−1+1).
Therefore, the rth moment of the BEPL distribution from equation (2.6) is given by µ0r = αβ 2ω B(a, b)(β+ 1) ∞ X i=0 b−1 i (−1)i × Z ∞ 0 xr(1 +xα)xα−1 1− 1 + βx α β+ 1 e−βxα ω(a+i)−1 e−βxαdx.
Now, using Lemma 3.1 withm =ω(a+i), we have
µ0r = αβ 2ω B(a, b)(β+ 1) ∞ X i=0 b−1 i (−1)iL1(α, β, ω(a+i), r). (3.3)
The mean, variance, coefficient of variation (CV), coefficient of skewness (CS) and coefficient of kurtosis (CK) are given by
µ=µ01 = αβ 2ω B(a, b)(β+ 1) ∞ X i=0 b−1 i (−1)iL1(α, β, ω(a+i),1), (3.4) σ2 =µ02−µ2, (3.5) CV = σ µ = p µ0 2−µ2 µ = s µ02 µ2 −1, (3.6) CS = E[(X−µ) 3] [E(X−µ)2]3/2 = µ03−3µµ02+ 2µ3 (µ0 2 −µ2)3/2 , (3.7) and CK = E[(X−µ) 4] [E(X−µ)2]2 = µ04 −4µµ03+ 6µ2µ0 2−3µ4 (µ0 2−µ2)2 , (3.8)
respectively.
Table 3.1 lists the first six moments of the BEPL distribution for selected values of the parameters by fixingα= 1.5, β= 1.0 and ω= 1.5.These values can be determined numerically using R and MATLAB. Algorithms to calcu-late the pdf moments, reliability, mean deviations, R´enyi entropy, maximum likelihood estimators and variance-covariance matrix of the BEPL distribution are provided in the appendix.
Table 3.1: Moments of the BEPL distribution for some parameter values;
α= 1.5, β= 1.0 and ω= 1.5 µ0s a= 0.5, b= 1.5 a = 1.5, b= 1.5 a= 1.5, b = 2.5 a= 2.5, b= 1.5 µ01 0.8348214 1.407729 1.129684 1.685774 µ02 1.035256 2.304562 1.472533 3.13659 µ03 1.608199 4.258207 2.149913 6.366501 µ04 2.928858 8.712316 3.450266 13.97437 µ05 6.037089 19.4759 6.007677 32.94413 µ06 13.77834 47.09621 11.24152 82.9509 Variance 0.33832923 0.322861063 0.19634706 0.294756021 Skewness 0.90987141 0.572390725 0.491885701 0.532003392 Kurtosis 3.760808259 3.40572536 3.236667109 3.434460209
3.2
Conditional Moments
For lifetime models, it is useful to know the conditional moments defined asE(Xr|X > x).In order to calculate the conditional moments, we consider the following lemma:
Lemma 3.2. Let L2(α, β, m, r, t) = Z ∞ t (1 +xα)xα+r−1 1− 1 + βx α β+ 1 e−βxα m−1 e−βxαdx. then L2(α, β, m, r, t) = ∞ X j=0 j X k=0 k+1 X l=0 m−1 j j k k+ 1 l (−1)jβkΓ(l+rα−1+ 1, β(j + 1)tα) α(β+ 1)j[β(j+ 1)](l+rα−1+1) ,
where Γ(a, t) = Rt∞xa−1s−xdx is the upper incomplete gamma function.
Proof. Using the same procedure that was used in Lemma 3.1, this can be
simplified into the following form.
L2(α, β, m, r, t) = ∞ X j=0 j X k=0 k+1 X l=0 m−1 j j k k+ 1 l (−1)jβk (β+ 1)j (3.9) × Z ∞ t xα+αl+r−1e(−jβxα−βxα)dx. (3.10) Now consider, R∞ t x α+αl+r−1e(−jβxα−βxα)
dx, and let u = β(j + 1)xα, then du dx =αβ(j+ 1)x α−1 andx= u β(j+ 1) 1/α
. The above integral can be rewrit-ten by using the complementary incomplete gamma function Γ(a, t) =Rt∞xa−1e−xdx.
Consequently, L2(α, β, m, r, t) = ∞ X j=0 j X k=0 k+1 X l=0 m−1 j j k k+ 1 l (−1)jβkΓ(l+rα−1+ 1, β(j + 1)tα) α(β+ 1)j[β(j+ 1)](l+rα−1+1) .
Now using Lemma 3.2, the rthconditional moment of the BEPL distribution
is given by E(Xr|X > x) = αβ 2ω B(a, b)(β+ 1) ∞ X i=0 b−1 i (−1)i L2(α, β, ω(a+i), r, x) 1−FBEP L(x;α, β, ω, a, b) = αβ 2ω (β+ 1) ∞ X i=0 b−1 i (−1)i L2(α, β, ω(a+i), r, x) B(a, b)−BGELP(x)(a, b) .
The mean residual lifetime function is given byE(X|X > x)−x.The moment generating function (MGF) of the BEPL distribution is given by
MX(t) = αβ2ω B(a, b)(β+ 1) ∞ X i=0 ∞ X n=0 b−1 i (−1)it n n!L1(α, β, ω(a+i), n).(3.11)
3.3
Reliability
We derive the reliabilityRwhenXandY have independent BEPL(α1, β1, ω1, a1, b1)
(2.2) that the BEPL cdf can be written as: FBEP L(x;α, β, ω, a, b) = 1 B(a, b) Z GEP L(x;α,β,ω) 0 ta−1(1−t)b−1dt, = 1 B(a, b) ∞ X j=0 b−1 j (−1)j a+j 1− 1 + βx α β+ 1 e−βxα ω(a+j) . (3.12) Now, from Equations (3.12) and (2.3), we obtain
R = P(X > Y) = Z ∞ 0 fX(x;α1, β1, ω1, a1, b1)FY(x;α2, β2, ω2, a2, b2)dx = Z ∞ 0 α1β12ω1 B(a1, b1)(β1+ 1) (1 +xα1)xα1−1e−β1xα1 × 1− 1 + β1x α1 β1+ 1 e−β1xα1 ω1a1−1 × 1− 1− 1 + β1x α1 β1+ 1 e−β1xα1 ω1b1−1 × 1 B(a2, b2) ∞ X j=0 b2−1 j (−1)j a2+j 1− 1 + β2x α2 β2 + 1 e−β2xα2 ω2(a2+j) dx. (3.13) We apply the following series representations:
1− 1 + β1x α1 β1+ 1 e−β1xα1 ω1a1−1 = ∞ X k=0 ω1a1−1 k (−1)k 1 + β1x α1 β1+ 1 k e−β1kxα1 = ∞ X k=0 k X m=0 ω1a1−1 k k m (−1)kβm 1 xmα1 (β1+ 1)m e−β1kxα1, (3.14) 1− 1− 1 + β1x α1 β1+ 1 e−β1xα1 ω1b1−1 = ∞ X l=0 ∞ X p=0 p X n=0 b1−1 l ω1l p p n × (−1) l+pβn 1xnα1 (β1+ 1)n e−β1pxα1 (3.15)
and 1− 1 + β2x α2 β2+ 1 e−β2xα2 ω2(a2+j) = ∞ X q=0 q X t=0 ω2(a2 +j) q q t (−1)qβt 2xtα2 (β2)t e−β2qxα2. (3.16) By substituting Equations (3.14), (3.15) and (3.16) into Equation (3.13), we
obtain R = Z ∞ 0 α1β12ω1 B(a1, b1)(β1 + 1) (1 +xα1)xα1−1e−β1xα1 × ∞ X k=0 k X m=0 ω1a1−1 k k m (−1)kβm 1 xmα1 (β1+ 1)m e−β1kxα1 × ∞ X l=0 ∞ X p=0 p X n=0 b1−1 l ω1l p p n (−1)l+pβn 1xnα1 (β1+ 1)n e−β1pxα1 × 1 B(a2, b2) ∞ X j=0 b2−1 j (−1)j a2+j × ∞ X q=0 q X t=0 ω2(a2+j) q q t (−1)qβt 2xtα2 (β2)t e−β2qxα2dx = α1ω1 B(a1, b1)B(a2, b2) ∞ X k,l,p,j,q=0 k X m=0 p X n=0 q X t=0 ω1a1−1 k k m b1−1 l ω1l p × p n b2 −1 j ω2(a2+j) q q t (−1)k+l+p+j+qβm+n+2 1 β2t (β1+ 1)m+n+1(β2+ 1)t(a2+j) × Z ∞ 0 (1 +xα1)x(m+n+1)α1+tα2−1exp(−[β 1(1 +p+k)xα1+β2qxα2])dx. (3.17) Note that, Z ∞ 0 (1 +xα1)x(m+n+1)α1+tα2−1 ×e−β1(1+p+k)xα1−β2qxα2dx = ∞ X s=0 ∞ X r=0 α1 r (−1)sβ1s(1 +p+k)s × Z ∞ 0 x(m+n+s+1)α1+tα2+r−1e−β2qxα2dx. (3.18)
Using the definition of gamma function, we have ∞ Z 0 x(m+n+s+1)α1+tα2+r−1e−β2qxα2dx= Γ((m+n+s+ 1)α1α −1 2 +rα −1 2 +t) α2(β2q)(m+n+s+1)α1α −1 2 +rα −1 2 +t . (3.19) Substituting Equation (3.19) into Equation (3.18), we obtain
Z ∞ 0 (1 +xα1)x(m+n+1)α1+tα2−1 ×e−β1(1+p+k)xα1−β2qxα2dx = ∞ X s=0 ∞ X r=0 α1 r (−1)sβ1s(1 +p+k)s × Γ((m+n+s+ 1)α1α −1 2 +rα −1 2 +t) α2(β2q)(m+n+s+1)α1α −1 2 +rα −1 2 +t . (3.20) Finally, substituting Equation (3.20) into (3.17), we obtain
R = α1ω1 B(a1, b1)B(a2, b2) ∞ X k,l,p,j,q=0 k X m=0 p X n=0 q X t=0 ω1a1−1 k k m b1−1 l ω1l p × p n b2 −1 j ω2(a2+j) q q t (−1)k+l+p+j+qβ1m+n+2β2t (β1+ 1)m+n+1(β2+ 1)t(a2+j) × ∞ X s=0 ∞ X r=0 α1 r (−1)sβ1s(1 +p+k)s × Γ((m+n+s+ 1)α1α −1 2 +rα −1 2 +t) α2(β2q)(m+n+s+1)α1α −1 2 +rα −1 2 +t . = α1ω1 B(a1, b1)B(a2, b2) ∞ X k,l,p,j,q,s,r=0 k X m=0 p X n=0 q X t=0 ω1a1−1 k k m b1−1 l ω1l p × p n b2−1 j ω2(a2+j) q q t α1 r (−1)k+l+p+j+q+sβm+n+s+2 1 β2t(1 +p+k)s (β1+ 1)m+n+1(β2+ 1)t(a2+j) × Γ((m+n+s+ 1)α1α −1 2 +rα −1 2 +t) α2(β2q)(m+n+s+1)α1α −1 2 +rα −1 2 +t .
4
Mean Deviations, Bonferroni and Lorenz Curves
In this section, we present the mean deviation about the mean, the mean deviation about the median, Bonferroni and Lorenz curves. Bonferroni and
Lorenz curves are income inequality measures that are also useful and appli-cable in other areas including reliability, demography, medicine and insurance. The mean deviation about the mean and mean deviation about the median are defined by D(µ) = Z ∞ 0 |x−µ|f(x)dx and D(M) = Z ∞ 0 |x−M |f(x)dx,
respectively, whereµ=E(X) andM =M edian(X) =F−1(1/2) is the median
ofF.These measuresD(µ) andD(M) can be calculated using the relationships:
D(µ) = 2µF(µ)−2µ+ 2 Z ∞ µ xf(x)dx= 2µF(µ)−2 Z µ 0 xf(x)dx, and D(M) =−µ+ 2 Z ∞ M xf(x)dx=µ−2 Z M 0 xf(x)dx.
Now using Lemma 3.2, we have
D(µ) = 2µFBEP L(µ)−2µ+ 2αβ2ω B(a, b)(β+ 1) ∞ X i=0 b−1 i (−1)iL2(α, β, ω(a+i),1, µ) and D(M) = −µ+ 2αβ 2ω B(a, b)(β+ 1) ∞ X i=0 b−1 i (−1)iL2(α, β, ω(a+i),1, M).
Lorenz and Bonferroni curves are given by
L(FBEP L(x)) = Rx 0 xfBEP L(x)dx E(X) , and B(FBEP L(x)) = L(FBEP L(x)) FBEP L(x) , or L(p) = 1 µ Z q 0 xfBEP L(x)dx, and B(p) = 1 pµ Z q 0 xfBEP L(x)dx, respectively, where q = F−1
BEP L(p). Using Lemma 3.2, we can re-write Lorenz
and Bonferroni curves as
B(p) = 1 pµ Z q 0 xfBEP L(x)dx = 1 pµ Z ∞ 0 xfBEP L(x)dx− Z ∞ q xfBEP L(x)dx = 1 pµ " µ− αβ 2ω B(a, b)(β+ 1) ∞ X i=0 b−1 i (−1)iL2(α, β, ω(a+i),1, q) # ,
and L(p) = 1 µ Z q 0 xfBEP L(x)dx = 1 µ Z ∞ 0 xfBEP L(x)dx− Z ∞ q xfBEP L(x)dx = 1 µ " µ− αβ 2ω B(a, b)(β+ 1) ∞ X i=0 b−1 i (−1)iL2(α, β, ω(a+i),1, q) # .
5
Order Statistics and Measures of Uncertainty
In this section, we present distribution of order statistics, Shannon entropy [23], [24], as well as R´enyi entropy [25] for the BEPL distribution. The concept of entropy plays a vital role in information theory. The entropy of a random variable is defined in terms of its probability distribution and is a good measure of randomness or uncertainty.
5.1
Distribution of Order Statistics
Order Statistics play an important role in probability and statistics. In this section, we present the distribution of the order statistics for the BEPL distribution. Suppose thatX1, X2, . . . , Xn is a random sample of sizenfrom a
continuous pdf, f(x). Let X1:n< X2:n< . . . < Xn:n denote the corresponding
order statistics. IfX1, X2, . . . , Xnis a random sample from BEPL distribution,
it follows from Equations (2.2) and (2.3) that the pdf of thekthorder statistic, say Yk =Xk:n is given by fk(yk) = n! (k−1)!(n−k)!fBEP L(yk) n−k X l=0 n−k l (−1)l BGEP L(yk;α,β,ω)(a, b) B(a, b) k+l−1 × αβ 2ω B(a, b)(β+ 1)(1 +y α k)y α−1 k exp(−βy α k) [V(yk)] ωa−1 [1−Vω(yk))] b−1 = αβ 2ωn!(1 +yα k)y α−1 k exp(−βykα) (β+ 1)(k−1)!(n−k)! n−k X l=0 b−1 X m=0 n−k l b−1 m × (−1) l+m (B(a, b))k+l−1 BGEP L(yk;α,β,ω)(a, b) k+l−1 [V(yk)]ω(a+m) −1 ,
whereV(yk) =GP L(yk;α, β, ω) = 1− 1 + βy α k β+ 1
exp(−βykα) andGEP L(yk;α, β, ω) =
Vω(yk). The corresponding cdf of Yk is Fk(yk) = n X j=k n−j X l=0 n j n−j l (−1)l[FBEP L(yk)] j+l = n X j=k n−j X l=0 n j n−j l (−1)l BGEP L(yk;α,β,ω)(a, b) B(a, b) j+l = n X j=k n−j X l=0 n j n−j l (−1)l [B(a, b)]j+l BGEP L(yk;α,β,ω)(a, b) j+l .
5.2
Shannon Entropy
Shannon entropy [23],[24] is defined byH[fBEP L] =E[−log(fBEP L(X;α, β, ω, a, b))].
Thus, we have H[fBEP L] = log B(a, b)(β+ 1) αβ2ω −E[log(1 +Xα)] − (α−1)E[log(X)] +βE[Xα] − (ωa−1)E log 1− 1 + βX α 1 +β e−βXα − (b−1)E log 1− 1− 1 + βX α 1 +β e−βXα ω . (5.1) Note that, for|x|<1, using the series representation log(1+x) =P∞q=1 (−1)qq+1xq,
we obtain E[log(1 +Xα)] =− ∞ X q=1 (−1)q q E[X qα], (5.2) E[log(X)] = − ∞ X p=1 p X s=0 p s (−1)s p E[X s], (5.3) E log 1− 1 + βX α 1 +β e−βXα = − ∞ X t=1 t X u=0 t u βu t(β+ 1)u × EXuαe−βtXα (5.4)
and E log 1− 1− 1 + βX α 1 +β e−βXα ω = ∞ X c=1 ∞ X d=0 d X e=0 ωc d d e (−1)d+1βe c(β+ 1)e × EXeαe−βdXα. (5.5) By using the results in Lemma 3.1, we can calculate Equations (5.2),(5.3),(5.4) and (5.5).
Now, we obtain Shannon entropy for the BEPL distribution as follows:
H[fBEP L] = log B(a, b)(β+ 1) αβ2ω + αβ 2ω B(a, b)(β+ 1) ∞ X i=1 b−1 i (−1)i × ∞ X q=1 (−1)q q L1(α, β, ω(a+i), qα) + (α−1) ∞ X p=1 ∞ X s=0 p s (−1)s p L1(α, β, ω(a+i), s) + βL1(α, β, ω(a+i), α) + (ωa−1) ∞ X t=1 ∞ X v=0 ∞ X u=0 t u βu+vtv−1(−1)v (β+ 1)u+1v! × L1(α, β, ω(a+i), α(u+v)) + (b−1) ∞ X c=1 ∞ X d=0 d X e=0 ∞ X f=0 ωc d d e βe+fdf(−1)d+f c(β+ 1)e+1f! × L1(α, β, ω(a+i), α(e+f)) . (5.6)
5.3
R´
enyi Entropy
R´enyi entropy [25] is an extension of Shannon entropy. R´enyi entropy is defined to be IR(v) = 1 1−v log Z ∞ 0 fBEP Lv (x;α, β, ω, a, b)dx , v 6= 1, v >0. (5.7) R´enyi entropy tends to Shannon entropy as v → 1. Note that by using the
series expansion in Equation (3.1),and Equation (2.3), we have Z ∞ 0 fBEP Lv (x;α, β, ω, a, b)dx = αβ2ω B(a, b)(β+ 1) v ∞ X i,j,p,q=0 ∞ X k=0 q X r=0 v i ωav−v j × j k bv−v p ωp q q r (−1)j+p+qβk+r (β+ 1)k+r) × Z ∞ 0 xα(i+k+r+v)−ve−β(v+j+q)xαdx.
Now letu=β(v+j+q)xα, then
∞ Z 0 xα(i+k+r+v)−ve−β(v+j+q)xαdx= Γ(i+k+r+v− (v−1) α ) α[β(v+j+q)]i+k+r+v− (v−1) α ) .
Consequently, R´enyi entropy is given by
IR(v) = 1 1−v log αβ2ω B(a, b)(β+ 1) v ∞ X i,j,p,q=0 ∞ X k=0 q X r=0 v i ωav−v j j k × bv−v p ωp q q r (−1)j+p+qβk+r (β+ 1)k+r × Γ(i+k+r+v− (v−1) α ) α[β(v+j+q)]i+k+r+v−(v −1) α , (5.8) forv 6= 1, v >0.
5.4
s-Entropy
The s-entropy for the BEPL distribution is defined by
Hs[fBEP L(X;α, β, ω, a, b)] = 1 s−1 1− Z ∞ 0 fBEP Ls (x;α, β, ω, a, b)dx if s 6= 1, s >0, E[−logf(X)] if s= 1.
have Z ∞ 0 fBEP Ls (x;α, β, ω, a, b)dx = αβ2ω B(a, b)(β+ 1) s ∞ X i,j,p,q=0 ∞ X k=0 q X r=0 s i × ωas−s j j k bs−s p ωp q q r × (−1) j+p+qβk+r (β+ 1)k+r Γ(i+k+r+s− (s−α1)) α[β(s+j+q)]i+k+r+s− (s−1) α .
Consequently, s-entropy is given by
Hs[fBEP L(X;α, β, ω, a, b)] = 1 s−1 − 1 s−1 αβ2ω B(a, b)(β+ 1) s × ∞ X i,j,p,q=0 ∞ X k=0 q X r=0 s i ωas−s j j k × bs−s p ωp q q r (−1)j+p+qβk+r (β+ 1)k+r × Γ(i+k+r+s− (s−1) α ) α[β(s+j+q)]i+k+r+s−(s −1) α fors 6= 1, s >0.
6
Maximum Likelihood Estimation
In this section, the maximum likelihood estimates of the BEPL parameters
α, β, ω, aand b are presented. Ifx1, x2, . . . , xn is a random sample from BEPL
distribution, the log-likelihood function is given by logL(α, β, ω, a, b) = nlog αβ2ω B(a, b)(β+ 1) + n X i=1 log(1 +xαi) + (α−1) n X i=1 log(xi)−β n X i=1 xαi + (ωa−1) n X i=1 logV(xi) + (b−1) n X i=1 log [1−Vω(xi)], where V(xi) = GP L(xi;α, β) = 1− 1 + βx α i β+ 1
exp (−βxαi). The partial derivatives of logL(α, β, ω, a, b) with respect to the parameters a, b, α, β and
ω are: ∂logL(α, β, ω, a, b) ∂a = n[ψ(a+b)−ψ(a)] +ω n X i=1 logV(xi), ∂logL(α, β, ω, a, b) ∂b = n[ψ(a+b)−ψ(b)] + n X i=1 log [1−Vω(xi)], ∂logL(α, β, ω, a, b) ∂α = n α + n X i=1 log(xi) xαi 1 +xα i −βxαi + 1 − (ωa−1) n X i=1 ∂V(xi)/∂α V(xi) + ω(b−1) n X i=1 [V(xi)]ω −1 ∂V(xi)/∂α 1−Vω(x i) , ∂logL(α, β, ω, a, b) ∂β = n(β+ 2) β(β+ 1) − n X i=1 xαi −(ωa−1) n X i=1 ∂V(xi)/∂β V(xi) + ω(b−1) n X i=1 [V(xi)]ω −1 ∂V(xi)/∂β 1−Vω(x i) and ∂logL(α, β, ω, a, b) ∂ω = n ω −(b−1) n X i=1 Vω(x i) logV(xi) 1−Vω(x i) +a n X i=1 logV(xi), respectively, where ∂V(xi) ∂α = β2 β+ 1log(xi)(1 +x α i)xαi exp(−βxαi) and ∂V(xi) ∂β = 1 + βx α i β+ 1 − 1 (β+ 1)2 xαi exp(−βxαi).
When all the parameters are unknown, numerical methods must be applied to determine the estimates of the model parameters since the system of equations is not in closed form. Therefore, the maximum likelihood estimates, ˆΘ of Θ = (α, β, ω, a, b) can be determined using an iterative method such as the Newton-Raphson procedure.
6.1
Fisher Information Matrix
In this section, we present a measure for the amount of information. This information can be used to obtain bounds on the variance of estimators and as well as approximate the sampling distribution of an estimator obtained from a large sample. Moreover, it can be used to obtain an approximate confidence interval in the case of a large sample.
Let X be a random variable with the BEPL pdf fBEP L(·; Θ), where Θ =
(θ1, θ2, θ3, θ4, θ5)T = (α, β, ω, a, b)T. Then, Fisher information matrix (FIM) is
the 5×5 symmetric matrix with elements: Iij(Θ) =EΘ ∂log(fBEP L(X; Θ)) ∂θi ∂log(fBEP L(X; Θ)) ∂θj .
If the density fBEP L(·; Θ) has a second derivative for all i and j, then an
alternative expression for Iij(Θ) is
Iij(Θ) =EΘ ∂2log(f BEP L(X; Θ)) ∂θi∂θj .
For the BEPL distribution, all second derivatives exist; therefore, the formula above is appropriate and most importantly significantly simplifies the compu-tations. Elements of the FIM can be numerically obtained by MATLAB or MAPLE software. The total FIMIn(Θ) can be approximated by
Jn( ˆΘ)≈ −∂ 2logL ∂θi∂θj Θ= ˆΘ |. 5×5 (6.1) For real data, the matrix given in Equation (6.1) is obtained after the conver-gence of the Newton-Raphson procedure in MATLAB or R software.
6.2
Asymptotic Confidence Intervals
In this section, we present the asymptotic confidence intervals for the pa-rameters of the BEPL distribution. The expectations in the Fisher Informa-tion Matrix (FIM) can be obtained numerically. Let ˆΘ = ( ˆα,β,ˆ ω,ˆ ˆa,ˆb) be the maximum likelihood estimate of Θ = (α, β, ω, a, b).Under the usual regularity conditions and that the parameters are in the interior of the parameter space, but not on the boundary, we have: √n( ˆΘ− Θ) −→d N5(0,I−1(Θ)), where
I(Θ) is the expected Fisher information matrix. The asymptotic behavior is still valid if I(Θ) is replaced by the observed information matrix evaluated at ˆΘ, that is J( ˆΘ). The multivariate normal distribution with mean vector 0 = (0,0,0,0,0)T and covariance matrixI−1(Θ) can be used to construct confi-dence intervals for the model parameters. That is, the approximate 100(1−η)% two-sided confidence intervals for α, β, ω, aand b are given by
ˆ α±Zη/2 q I−1 αα( ˆΘ), βˆ±Zη/2 q I−ββ1( ˆΘ), ωˆ±Zη/2 q I−1 ωω( ˆΘ), ˆ a±Zη/2 q I−1 aa( ˆΘ) and ˆb±Zη/2 q I−bb1( ˆΘ)
respectively, whereI−αα1( ˆΘ),Iββ−1( ˆΘ),Iωω−1( ˆΘ),I−aa1( ˆΘ) and I−bb1( ˆΘ) are diagonal el-ements of I−n1( ˆΘ) = (nIΘ))ˆ −1 and Zη/2 is the upper (η/2)th percentile of a
standard normal distribution.
We can use the likelihood ratio (LR) test to compare the fit of the BEPL distribution with its sub-models for a given data set. For example, to test
α =ω = 1, the LR statistic is ω∗ = 2[ln(L(ˆa,ˆb,β,ˆ α,ˆ ωˆ))−ln(L(˜a,˜b,β,˜ 1,1))],
where ˆa, ˆb, β,ˆ αˆ and ˆω are the unrestricted estimates, and ˜a, ˜b, and ˜β are the restricted estimates. The LR test rejects the null hypothesis if δ∗ > χ2,
where χ2 denote the upper 100% point of the χ2 distribution with 2 degrees of freedom.
7
Applications
In this section, the BEPL distribution is applied to real data in order to illustrate the usefulness and applicability of the model. We fit the density functions of the beta-exponentiated power Lindley (BEPL), beta exponen-tiated Lindley (BEL), exponenexponen-tiated power Lindley (EPL) [4], beta power Lindley (BPL), power Lindley (PL), and Lindley (L) distributions. We pro-vide examples to illustrate the flexibility of the BEPL distribution in contrast to other models including the BEL, BPL, PL, L, beta-Weibull (BW) [26], beta-exponential (BE) [15] and Weibull (W) distributions for data modeling purposes. The pdf of the BW distribution [26] is given by
fBW(x;α, λ, a, b) =
αλα
B(a, b)x
for x >0, α > 0, λ > 0, a > 0, b > 0. When α = 1, the beta exponential pdf is obtained, [15].
Estimates of the parameters of the distributions, standard errors (in paren-theses), Akaike Information Criterion (AIC = 2p−2 log(L)), Consistent Akaike Information Criterion (AICC =AIC+ 2np−(pp+1)−1), Bayesian Information Crite-rion (BIC =plog(n)−2 log(L)), whereL=L( ˆΘ) is the value of the likelihood function evaluated at the parameter estimates,nis the number of observations, and pis the number of estimated parameters are obtained.
The first data set represents the maintenance data with 46 observations re-ported on active repair times (hours) for an airborne communication transceiver discussed by Alven [27], Chhikara and Folks [28] and Dimitrakopoulou et al. [29]. It consists of the observations listed below: 0.2, 0.3, 0.5, 0.5, 0.5, 0.5, 0.6, 0.6, 0.7, 0.7, 0.7, 0.8, 0.8, 1.0, 1.0, 1.0, 1.0, 1.1, 1.3, 1.5, 1.5, 1.5, 1.5, 2.0, 2.0, 2.2, 2.5, 2.7, 3.0, 3.0, 3.3, 3.3, 4.0, 4.0, 4.5, 4.7, 5.0, 5.4, 5.4, 7.0, 7.5, 8.8, 9.0, 10.3, 22.0, 24.5.
The second data set represents the remission times (in months) of a random sample of 128 bladder cancer patients reported in Lee and Wang ([30]). See the table below.
Table 7.1: Cancer Patients Data, Lee and Wang [30]
0.08 2.09 3.48 4.87 6.94 8.66 13.11 23.63 0.20 2.23 3.52 4.98 6.97 9.02 13.29 0.40 2.26 3.57 5.06 7.09 9.22 13.80 25.74 0.50 2.46 3.64 5.09 7.26 9.47 14.24 25.82 0.51 2.54 3.70 5.17 7.28 9.74 14.76 26.31 0.81 2.62 3.82 5.32 7.32 10.06 14.77 32.15 2.64 3.88 5.32 7.39 10.34 14.83 34.26 0.90 2.69 4.18 5.34 7.59 10.66 15.96 36.66 1.05 2.69 4.23 5.41 7.62 10.75 16.62 43.01 1.19 2.75 4.26 5.41 7.63 17.12 46.12 1.26 2.83 4.33 5.49 7.66 11.25 17.14 79.05 1.35 2.87 5.62 7.87 11.64 17.36 1.40 3.02 4.34 5.71 7.93 11.79 18.10 1.46 4.40 5.85 8.26 11.98 19.13 1.76 3.25 4.50 6.25 8.37 12.02 2.02 3.31 4.51 6.54 8.53 12.03 20.28 2.02 3.36 6.76 12.07 21.73 2.07 3.36 6.93 8.65 12.63 22.69 - - -
-The third data set consists of the number of successive failures for the air conditioning system of each member in a fleet of 13 Boeing 720 jet airplanes (Proschan [31]). The data is presented in Table 7.2.
Table 7.2: Air conditioning system data 194 413 90 74 55 23 97 50 359 50 130 487 57 102 15 14 10 57 320 261 51 44 9 254 493 33 18 209 41 58 60 48 56 87 11 102 12 5 14 14 29 37 186 29 104 7 4 72 270 283 7 61 100 61 502 220 120 141 22 603 35 98 54 100 11 181 65 49 12 239 14 18 39 3 12 5 32 9 438 43 134 184 20 386 182 71 80 188 230 152 5 36 79 59 33 246 1 79 3 27 201 84 27 156 21 16 88 130 14 118 44 15 42 106 46 230 26 59 153 104 20 206 5 66 34 29 26 35 5 82 31 118 326 12 54 36 34 18 25 120 31 22 18 216 139 67 310 3 46 210 57 76 14 111 97 62 39 30 7 44 11 63 23 22 23 14 18 13 34 16 18 130 90 163 208 1 24 70 16 101 52 208 95 62 11 191 14 71 - - -
-Estimates of the parameters of BEPL distribution (standard error in paren-theses), Akaike Information Criterion, Consistent Akaike Information Criterion and Bayesian Information Criterion are given in Table 7.3 for the active repair time data, in Table 7.4 for the cancer patient data and in Table 7.5 for the air conditioning system data.
Table 7.3: Estimates of Models for Repair Times Data
Estimates Statistics
Model α β ω a b −2 logL AIC AICC BIC
BEPL(α, β, ω, a, b) 0.08792 7.5983 69.3171 41.0765 2.1953 199.3 209.3 210.8 218.5 (0.2992) (15.9158) (1260.74) (15.5515) (6.2258) PL(α, β,1,1,1) 0.7581 0.6757 1 1 1 210.0 214.0 214.3 217.7 (0.07424) (0.1016) L(1, β,1,1,1) 1 0.4664 1 1 1 220.0 222.0 222.1 223.8 (0.0499) BL(1, β,1, a, b) 1 1.6145 1 0.9513 0.2007 212.9 218.9 219.5 224.4 (0.03449) (0.2505) (0.03284) BW(α, β,−,a, b) 0.5408 36.6023 - 41.4065 0.1263 198.0 206.0 207.0 213.3 (0.1821) (21.0749) (3.4654) (0.1214) W(α, β,−,1, 1) 0.8986 0.2949 - 1 1 208.9 212.9 213.2 216.6 (0.09576) (0.05138) BE(1, β,−,a, b) 1 0.01218 - 0.9322 21.2530 209.9 215.9 216.4 221.3 (0.003109) (0.1793) (1.7710)
For the repair times data set, the LR statistic for the hypothesis H0:
P L(α, β,1,1,1) against Ha: BEP L(α, β, ω, a, b), is ω∗ = 10.7. The p-value is
0.01346379<0.05. Therefore, there is a significant difference between PL and BEPL distributions. A LR test ofH0: L(1, β,1,1,1) vsHa:BEP L(α, β, ω, a, b)
a significant difference between L and BEPL distributions. There is also a significant difference between PL and L distributions where ω∗ = 10.0 with a p-value of 0.00107136 < 0.01. Moreover, the values of the statistics AIC and AICC are smaller for the BEPL distribution and show that the BEPL distri-bution is a “better” fit than its sub-models for the repair times data, however a comparison of BEPL and BW distributions shows that the four parameter BW distribution is slightly better.
The asymptotic covariance matrix of MLEs for BEPL model parameters, which is the FIM I−n1( ˆΘ), is given by
0.08952 −4.5464 −370.36 4.2174 −1.6999 −4.5464 253.31 19939 −184.68 74.4387 −370.36 19939 1589474 −15981 6439.33 4.2174 −184.68 −15981 241.85 −96.1693 −1.6999 74.4387 6439.33 −96.1693 38.761
and the 95% two-sided asymptotic confidence intervals for α, β, ω, aand b are given by 0.08792±0.586432,7.5983±31.194968,69.3171±2471.0504,41.0765±
30.48094 and 2.1953±12.202568, respectively. Plots of the fitted densities and the histogram of the repair time data are given in Figure 7.1.
Figure 7.1: Plot of the fitted densities for the Repair Times Data
Table 7.4: Estimates of Models for Cancer Patient Data
Estimates Statistics
Model α β ω a b −2 logL AIC AICC BIC BEPL(α, β, ω, a, b) 0.9049 0.3352 34.3398 0.0358 0.3598 818.8736 828.8736 829.3659 843.1337 (0.2657) (0.2508) (0.0039) (0.0199) (0.2251) BPL(α, β,1, a, b) 0.60245 0.8686 1 2.5744 0.7605 820.8393 828.8393 829.1645 840.2474 (0.2299) (0.4169) - (1.5238) (1.1546) PL(α, β,1,1,1) 0.8302 0.2943 1 1 1 826.7076 830.7076 830.8636 836.4117 (0.0472) (0.0370) L(1, β,1,1,1) 1 0.19614 1 1 1 839.0596 841.0596 841.0916 843.9118 (0.0499) BW(α, β,−,a, b) 0.6689 0.3304 - 2.7257 0.8808 821.3575 829.3575 829.6827 840.7657 (0.2368) (0.4177) (1.5572) (1.3743) W(α, β,−,1, 1) 1.0479 0.1046 - 1 1 828.1738 832.1738 832.2698 837.8778 (0.0676) (0.0093)
For the cancer patients data, the LR statistics for the test of the hypotheses
H0 : P L(α, β,1,1,1) againstHa : BEP L(α, β, ω, a, b) and H0 : L(1, β,1,1,1)
20.186 (p−value = 0.000459 < 0.001), respectively. Consequently, we re-ject the null hypothesis in favor of the BEPL distribution and conclude that the BEPL distribution is significantly better than the PL and L distributions. However, there is no significant difference between the BPL and BEPL distri-butions based on the LR test. Also, based on the values of the statistics: AIC, AICC and BIC, we conclude that the BPL distribution is the better fit for the cancer patient data. The BPL distribution is also slightly better that the BW distribution based on the values of these statistics. Plots of the fitted densities and the histogram for the cancer patient data are given in Figure 7.2.
Table 7.5: Estimates of Models for Air Conditioning System Data
Estimates Statistics
Model α β ω a b −2 logL AIC AICC BIC
BEPL(α, β, ω, a, b) 0.7945 0.1509 6.7278 0.2035 0.2303 2064.1 2074.1 2074.4 2090.2 (0.2706) (0.2102) (3.4546) (0.2146) (0.1512) BPL(α, β,1, a, b) 0.4316 0.4867 1 3.1251 0.9630 2066.7 2074.7 2074.9 2087.6 (0.0573) (0.1658) (0.4284) (0.8737) BEL(1, β, ω, a, b) 1 0.0453 7.5488 0.1048 0.2034 2064.8 2072.8 2073.0 2085.8 (0.0194) (3.9156) (0.0623) (0.0649) BL(1, β,1, a, b) 1 0.02343 1 0.4842 0.5378 2080.6 2086.6 2086.7 2096.3 (0.00972) (0.0538) (0.2302) PL(α, β,1,1,1) 0.6609 0.1807 1 1 1 2071.4 2075.4 2075.5 2081.9 (0.0316) (0.0165) L(1, β,1,1,1) 1 0.0215 1 1 1 2165.3 2167.3 2167.3 2170.5 (0.00111) BW(α, β,−,a, b) 0.7383 0.2719 - 2.7250 0.1188 2064.6 2072.6 2078.8 2085.6 (0.1114) (0.7861) (4.7308) (0.2421) W(α, β,−,1, 1) 0.9109 0.0114 - 1 1 2073.5 2077.5 2077.6 2084.0 (0.0504) (0.00097) BE(1, β,−,a, b) 1 0.00129 - 0.9048 7.6602 2075.2 2081.2 2081.4 2090.9 (0.000184) (0.0864) (0.3651)
For the air conditioning system data, the LR statistics for the test of the hypothesesH0 : BL(1, β,1, a, b) against Ha : BEP L(α, β, ω, a, b) is 16.5 (p−
value = 0.000263 < 0.001.) Consequently, we reject the null hypothesis in favor of the BEPL distribution and conclude that the BEPL distribution is significantly better than the BL distribution. The LR test statistics for the test of the hypotheses H0 : BL(1, β,1, a, b) against Ha : BEL(1, β, ω, a, b)
is 15.8 (p− value = 0.000704 < 0.001), so that the null hypothesis of BL model is rejected in favor of the alternative hypothesis of BEL model. The BPL distribution is also significantly better than the PL and BL models based on the LR test. However, there is no significant difference between the BPL and BEPL distributions, as well as between the BEL and BEPL distributions based on the LR test. The sub-models: BPL and BEL are better fits than the BEPL distribution for the air conditioning system data. Also, the values of the statistics: AIC, AICC and BIC, points to the BEL distribution, so we conclude that the BEL distribution is the better fit for the air conditioning system data. The BEL distribution also compares favorably with the BW distribution based on the values of these statistics. Plots of the fitted densities and the histogram for the air conditioning system data are given in Figure 7.3. Based on the values of these statistics, we conclude that the BEPL distri-bution and its sub-models can provide good fits for lifetime data. In the first data set, the BEPL distribution performed better than the BL, PL, L, BE, and Weibull distributions. The four parameter BW distribution was slightly better based on the values of AIC, AICC and BIC. In the second data set,
the BPL distribution performed better than the other models including the beta Weibull distribution. In the third data set, the BEL distribution as well as the BPL distribution seem to be the better fits, and the BEL distribution compares favorably with the BW distribution. The BEPL and its sub-models including the BEL and BPL distributions can provide better fits than other common lifetime models.
Figure 7.3: Plot of the fitted densities for the Air Conditioning System Data
8
Concluding Remarks
We have developed and presented the mathematical properties of a new class of distributions called the beta-exponentiated power Lindley (BEPL) distribution including the hazard and reverse hazard functions, monotonicity properties, moments, conditional moments, reliability, entropies, mean devia-tions, Lorenz and Bonferroni curves, distribution of order statistics, and
max-imum likelihood estimates. Applications of the proposed model to real data in order to demonstrate the usefulness of the distribution are also presented.
ACKNOWLEDGEMENTS.The authors are grateful to the referees for
some useful comments on an earlier version of this manuscript which led to this improved version.
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R Algorithms
#Define the pdf of BEPL
f1=function(x,alpha,beta,omega,a,b){y=(alpha*beta^2*omega*(1+x ^alpha) *x ^(alpha-1)*exp(-beta*x^alpha) *(1-(1+beta*x ^alpha /(1+beta))
*exp(-beta *x ^alpha)) ^(omega*a-1)) *(1-(1-(1+beta*x ^alpha /(1+beta)) *exp(-beta *x ^alpha))^omega) ^(b-1) /(beta(a,b)*(beta+1))
return(y) }
#Define the cdf of BEPL
F1=function(x,alpha,beta,omega,a,b){
y=pbeta((1-(1+beta*x^alpha/(1+beta))*exp(-beta*x^alpha))^omega,a,b) return(y)
}
#Define the moments of BEPL
moment=function(alpha,beta,omega,a,b,r){ f=function(x,alpha,beta,omega,a,b,r) {(x^r)*(f1(x,alpha,beta,omega,a,b))} y=integrate(f,lower=0,upper=Inf,subdivisions=100 ,alpha=alpha,beta=beta,omega=omega,a=a,b=b,r=r) return(y) }
#Define the reliability of BEPL reliability=function(alpha1,beta1,omega1,a1,b1,alpha2, beta2,omega2,a2,b2){ f=function(x,alpha1,beta1,omega1,a1,b1,alpha2,beta2,omega2,a2,b2) {f1(x,alpha1,beta1,omega1,a1,b1)*(F1(x,alpha2,beta2,omega2,a2,b2))} y=integrate(f,lower=0,upper=Inf,subdivisions=100,alpha1=alpha1, beta1=beta1,omega1=omega1,a1=a1,b1=b1,alpha2=alpha2, beta2=beta2,omega2=omega2,a2=a2,b2=b2) return(y) }
#Define Mean Deviation about the mean of BEPL delta1=function(alpha,beta,omega,a,b){ mu=moment(alpha,beta,omega,a,b,1)$ value f=function(x,alpha,beta,omega,a,b){(abs(x-mu)*f1(x.alpha,beta,omega,a,b)} y=integrate(f,lower=0,upper=Inf,subdivisions=100 ,alpha=alpha,beta=beta,omega=omega,a=a,b=b) return(y) }
#Define Mean Deviation about the median of BEPL delta2=function(alpha,beta,omega,a,b){
M=median(c(X)) #X is the data set
f=function(x,alpha,beta,omega,a,b){(abs(x-M)*f1(x.alpha,beta,omega,a,b)} y=integrate(f,lower=0,upper=Inf,subdivisions=100
,alpha=alpha,beta=beta,omega=omega,a=a,b=b) return(y)
Define the Renyi entropy of BEPL t=function(alpha,beta,omega,a,b,gamma){ f=function(x,alpha,beta,omega,a,b,gamma) {(f1(x,alpha,beta,omega,a,b))^(gamma)} y=integrate(f,lower=0,upper=Inf,subdivisions=100 ,alpha=alpha,beta=beta,omega=omega,a=a,b=b,gamma=gamma)$ value return(y) } Renyi=function(alpha,beta,omega,a,b,gamma){ y=log(t(alpha,beta,omega,a,b,gamma))/(1-gamma) return(y) }
#Calculate the maximum likelihood estimators and variance-covariance matrx of the BEPL
library('bbmle');
xvec<-c(X) #X is the data set
fn1<-function(alpha,beta,omega,a,b){ -sum(log(alpha*beta^2*omega/(beta(a,b)*(beta+1)))+log(1+xvec^alpha) +(alpha-1)*log(xvec)-beta*xvec^alpha+(omega*a-1) *log(1-(1+beta*xvec^alpha/(beta+1)) *exp(-beta*xvec^alpha))+(b-1)*log(1-(1-(1+beta*xvec^alpha/(beta+1)) *exp(-beta*xvec^alpha))^omega)) } mle.results1<-mle2(fn1,start=list(alpha=alpha,beta=beta, omega=omega,a=a,b=b),hessian.opt=TRUE) summary(mle.results1) vcov(mle.results1)
